This survey article looks at various subgroup properties that lie somewhere between the property of being a normal subgroup and the property of being a characteristic subgroup. The subgroup properties are organized according to different running themes.

Definitions

Normal subgroup

A subgroup of a group is said to be normal if it satisfies the following equivalent conditions:

It is invariant under all inner automorphisms. Thus, normality is the invariance property with respect to the property of an automorphism being inner. This definition also motivates the term invariant subgroup for normal subgroup (which was used earlier).

Relation between normality and characteristicity

Characteristic implies normal: A characteristic subgroup must be normal, since invariance under all automorphisms implies invariance under inner automorphisms.

Normal not implies characteristic: A normal subgroup need not be characteristic. For instance, in the group , both factors are normal but the coordinate exchange automorphism interchanges them, so neither is characteristic.

One notion of betweenness: invariance under the right kind of automorphisms

Normality is defined as the property of being invariant under all inner automorphisms, while characteristicity is defined as the property of being invariant under allautomorphisms. Thus, one way of looking for properties in between them is to look for invariance properties with respect to automorphism properties that are weaker than being an inner automorphism.

If is an automorphism property such that every inner automorphism of a group satisfies , then the property of being an -invariant subgroup is stronger than normality and weaker than characteristicity.

Automorphisms of certain orders

Suppose is a finite group. Then, , and hence, the order of the inner automorphism group of divides the order of . In particular, every inner automorphism of a group has order with no prime factors other than those of the order of .

We can look at the set of all elements of whose order has no prime factors other than those of . In other words, if is the set of prime factors of the order of , we are looking for the subgroup of generated by all the -automorphisms.